L(s) = 1 | + (−0.766 − 0.642i)3-s + (−0.766 + 1.32i)5-s + (0.173 + 0.300i)7-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)13-s + (1.43 − 0.524i)15-s − 1.87·17-s + (0.0603 − 0.342i)21-s + (−0.673 − 1.16i)25-s + (0.500 − 0.866i)27-s + (−0.5 + 0.866i)31-s − 0.532·35-s + 1.53·37-s + (0.939 − 0.342i)39-s + (−0.939 − 1.62i)43-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)3-s + (−0.766 + 1.32i)5-s + (0.173 + 0.300i)7-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)13-s + (1.43 − 0.524i)15-s − 1.87·17-s + (0.0603 − 0.342i)21-s + (−0.673 − 1.16i)25-s + (0.500 − 0.866i)27-s + (−0.5 + 0.866i)31-s − 0.532·35-s + 1.53·37-s + (0.939 − 0.342i)39-s + (−0.939 − 1.62i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09025218257\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09025218257\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 0.347T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.936638812272539007623976326103, −8.234166575522716072666202937586, −7.34027301904693615550156892369, −6.78984792652323577775899893505, −6.54456496789950237266869956467, −5.41664100408565716640796545973, −4.58032883635711901996472717360, −3.75896972628953209098223509902, −2.55678852059728142736891584574, −1.92695788602108619953694636416,
0.06019349182816634045108503560, 1.22952677383235132616985765277, 2.80871150141933921927577870020, 4.07772831943094740135274776692, 4.47549417812639687870051519295, 5.03293883619685911360967087803, 5.92213932929309779163457293428, 6.70072109287127429419603478457, 7.76556600650504329296894993653, 8.201751461680250141829090181963